The present invention relates to a method for establishing a common key within a group of subscribers using a publicly known mathematical group and a publicly known element of the group.
Encryption methods of varied types belong to state of the art and increasingly have commercial importance. They are used for sending messages over commonly accessible transmission media, but only the owners of a cryptokey are able to read these messages in plain text.
A known method for establishing a common key over unsecure communication channels is, for example, the method by W. Diffie and W. Hellmann (see DH-Method W. Diffie and M. Hellmann, see New Directions in Cryptography, IEEE Transaction on Information Theory, IT-22(6): 644-654, November 1976).
The basis of the Diffie Hellmann key exchange (DH-key exchange) is the fact that it is virtually impossible to compute logarithms modulo a large prime number p. In the example depicted below, Alice and Bob make use of this in that they each secretly select a number x or y, respectively, which are smaller than p (and relatively prime to p-1). Then, they (successively or simultaneously) send each other the xth (or yth) power modulo p of a publicly known number α. They are able to compute a common key K:=αxy mod p from the received powers by another exponentiation modulo p with x or y, respectively. An attacker who sees only αx mod p and αy mod p cannot compute K therefrom. (The only method for this which is known today would be to initially compute the logarithm, for example, of αx to base α modulo p, and to subsequently exponentiate αy therewith.)
